Lecture Notes in Computer Science- P19 doc

• Composite KP: if the knowledge of the KP may be divided into less KP reflecting knowledge contents, this KP is called as Composite KP.. 2.2 Relationships among KP The parent-child, th

• When adding new KP to the syllabus, the material reflecting this KP must be added into the courseware also

• When the KP does not longer present in the syllabus, any material reflecting this

KP must be deleted in the courseware Meanwhile, the KP reflected in the remain-ing material must be enough for the students to learn the syllabus

In order to deal with these cases, this paper discusses the relationships among KP and the relationships among material On the basis of them, it provides material Petri net (MPN) to validate some material property It is organized as follows Section 2 describes the KP net and clarifies the relationships among KP in the syllabus Based

on KP net, Section 3 presents the material net Section 4 introduces Petri net and gives the definition of MPN Then, it researchs some material property by the MPN In the section 5, it draws a conclusion and presents the future work

2 Knowledge Point

2.1 Overview of KP

The whole knowledge in a book consists of some KP, and these KP distribute from the various charpters It is essence that various material reflecting KP is learned in the e-learning In order to describe material, we must give some related conception of KP firstly Well-known, the KP is an teaching unit including knowledge in the learning process From the conception of KP, we know that KP is demarcated according to require-ment It brings forth a question about how to divide knowledge into KP properly? It is the basic principle of the knowledge demarcation that the partial completeness must

be ensured[4] For example, one chapter in the book may be a large KP The contents

of chapter may be divided into some sections also A section of in the chapter may be divided into some less KP again However, if KP is continuously divided, it would not reflect concrete knowledge contents lastly Hence, we have two conceptions:

• Atom KP: if the knowledge of the KP is further splitted, there would not exist any completeness knowledge contents This KP is called as Atom KP

• Composite KP: if the knowledge of the KP may be divided into less KP reflecting

knowledge contents, this KP is called as Composite KP If the KP C consists of the

KP A and KP B, it is called as the parent of KP A and KP B KP C is a Composite

KP The KP A and the KP B are called as children of KP C All of children of one

KP are called as brother KP each other

2.2 Relationships among KP

The parent-child, the association and the parallel are three relationships among KP:

• parent-child relationship: it is the relationship between a Composite KP and its children KP

• association relationship: it is the relationship that a KP may be learned directly after another KP have been mastered

• parallel relationship: if the relationship between two KP is not the parent-child relationship and the association relationship, it is a parallel relationship

In the association relationship among KP, if a KP must be mastered directly before another KP is learned, it is called as the Ancestor KP for another KP if a KP may be learned directly after another KP has been learned, it is called as the Descendant KP for another KP

2.3 KP Net Diagram

The relationships among KP are described with a net structure[4] The Composite KP can be divided into some little KP On the grounds of parent-child relationship, if a material reflects in an Atom KP, it will also reflect in the parent KP of this atom KP

So, on the discussion below, we assume that all KP should be atomistic

An arc is expressed as the association relationship among KP The arrowhead di-rects from the Ancestor KP to the Descendant KP If a KP has an arc pointing at it, this arc is called as In-arc for this KP If a KP has an arc backing at it, this arc is called as Out-arc for this KP

Definition 1 KP net: it is the net that is made up of the KP and the relationships

of them The node means the KP and the arc means the association relationship of the

KP

In an ordinary way, the KP net is andirected acyclic graph See figure 1

B

D E

C A

Fig 1 KP Net

At the figure 1, A, B, C, D and E means some Atom KP in the courseware The rela-tions of A and B, A and C, B and E, B and D, C and D are the association relarela-tionships

A is the ancestor B and C B and C are the descendant of A Likewise, D is descendant of

B and C B and C are the ancestor of D the relationship between B and C is the parallel,

and the relationship between D and E is the parallel also

3 Material

3.1 Material Set

Material is a media that can reflect some KP independently It may be a document,

picture, sound, etc The relationship between the material X and the KP A, B can be expressed as X={A,B} The expression X={A,B} means that the material X reflects the

KP set, which includes KP A and B

The relationship between the material and the KP is many to many One material may reflect with much KP, and one KP may be reflected with much material The material reflecting one KP can be learned only if learner masters the Ancestor KP of this KP Further, learner must master the Ancestor KP of this Ancestor KP Obvi-ously, the KP set, which includes the Descendant KP and the Ancestor KP, is redun-dant It might be simplified The algorithm is described as follows:

Step 1 selecting any element in the KP set described by material;

Step 2 searching all paths from starting KP to this KP;

Step 3 in the KP set reflected in the material, deleting all elements in the paths except for itself;

Step 4 selecting the remain element in the KP set, and repeating step 2 and step 3; After all elements in the KP set reflected in material are selected, the KP set, which only includes the surplus elements, is the simplest KP set of this material

Example 1: To figure 1, Supposed the KP sets reflected in some material as follows

X1={A}, X2={A}, X3={A,B}, X4={A,B,C,D}, X5={A,C}, X6={A,B,C}, X7={E} After

simplified, the KP sets is described as follows X1={A}, X2={A}, X3={B}, X4={D},

X5={C}, X6={B,C},X7={E}

Because there are much material reflecting the same KP set, we concern a material

set rather than single material

Definition 2 material set (MS): if all elements in a material set can replace with

each other when they reflect KP, this set can be called as a material set

About example 1, the MS are X12={X1,X2}, X3={X3}, etc

All material in the MS reflect the same KP set Below, we do not discuss the single material except for the MS

3.2 Relationship among Material Set

Because all material are equal when they reflect KP, there does not exist the parent-child relationship among material The association and the parallel are two relation-ships among MS

• association relationship: it is the relationship that the MS may be learned di-rectly after another MS have been mastered

• parallel relationship: if the relationship between two MS is not association rela-tionship, it is parallel relationship

In the association relationship, if a MS must be mastered directly beore another MS

is learned, it is called the Ancestor MS for another MS if a MS may be learned directly after another MS has been mastered, it is called the Descendant MS for another MS

It is relative about the Ancestor MS and the Descendant MS If MS A is the ances-tor of MS B, MS B is the descendant of MS A also

3.3 MS Net

An arc is expressed for the association relationship among MS The arrowhead directs from the Ancestor MS to the Descendant MS If a MS has an arc pointing at it, the arc

is called as In-arc for this MS If a MS has an arc backing at it, the arc is called as Out-arc for this MS The figure 2 is the MS net graph of the example 1

A X12

X6

B

BC C

Fig 2 MS Net

Definition 3 MS net: it is the net that is made up of the MS and the relationships among them The node means MS and the arc means the association relationship of MS

A MS may be ancestor and descendant of itself also MS net is an cyclic directed graph

At the figure 2, X12, X3, X4, X5, X6 and X7 means some MS in the courseware A,

B, C, D, E and BC means KP reflected in MS The relationships of X12 and X3, X12

and X5, X3 and X6, X5 and X6, X5 and X7, X3 and X4, X5 and X4, X6 and X4 are the association relationship X12 is the ancestor of X3 and X5 X3 and X5 are the descen-dant of X12 etc

4 The MS Specialty Research

4.1 Material Petri Net (MPN)

It is an effective system model verification tools for Petri net (PN)[9,10,11]

Definition 4 Petri net: it includes six items, PN=(P,T,F,K,W,M 0 )

P: is place set

T: is transition set (P , T≠∅, P∩T=∅)

F⊆(P×T) (T×P): is flow relation

K: defines the maximum token in a place

W: defines the weighted coefficient in token

M 0: is a start label

Definition 5 path: is a transition sequence σ=M 0 t 1 M 1 t 2 M 2 …t n M n in Petri net It

is for short σ =t 1 t 2 …t n, and called as trigger sequence of transition

Definition 6 accessibility: if it has a sequence transition t 1 , t 2 , …, t n from M 0 to

M n in PN, it can be say that M n is accessibility from M 0

Definition 7 Supposed N=(P,T,F,K,W,M 0 ) is Petri net N is MPN, if and only if

two conditions is satisfied as follows:

• there are two special places, which are Mstart and M stop M start place is the

beigin-ing, and M stop is the end

• There are a special transition tstop , all places can arrive at M stop only via t stop

Definition 8 material accessibility: if it has a sequence transition t 1 , t 2 , …, t stop

from M start to M stop in the MPN, it can be say that M stop is the material accessibility

from M start

4.2 Transformation from MS Net Graph to MPN

On the procession of the learning in the courseware, if the MS has been learned, which means that the KP reflected in this MS has been mastered, other KP in the

sequence may be learned continuously Therefore, the MS may be treat as place (P), and the KP may be treat as transition (T) The maximum of token (K) in the place is

the element numbers in the MS In order to describe the relationship of MS in MPN,

some definition must be introduced

• Stop KP set: it is a set where all KP had been learned

• Stop MS: it is a virtual MS included all KP that have been finished

• Void KP: is a virtual KP without any real knowledge the Void KP transits without fail

• Void MS: is a virtual MS without reflecting any real KP The learning of Void

MS does not need any premise

The transformation about four special relationships of MS are given through the figure 3

X3

X6

BC C

B X5

X5 A

X3 X12

(a)

(b)

tstop

E

stop

X3

X4

D BC

C

B

X5

(d) (c)

Fig 3 Four Special Relationship in the MPN